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Let be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on with square integrable initial condition is identically zero at all times whenever and the solution at a time are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.
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